AI 中文总结
该论文证明了Gram矩阵Hadamard幂相关的两个猜想，确定不等式取等条件，还提出其针对三个及以上向量的新推广。
AI 中文摘要
文献[Linear and Multilinear Algebra 74:1779--1798 (2026)]中考虑了针对向量$\boldsymbol{v}, \boldsymbol{w} \notin \boldsymbol{R}^n$的如下不等式：$\boldsymbol{\rVert}\boldsymbol{v}^p\boldsymbol{\rVert}\boldsymbol{\rVert}\boldsymbol{w}^p\boldsymbol{\rVert} - \boldsymbol{\rangle}\boldsymbol{v}^p,\boldsymbol{w}^p\boldsymbol{\rangle} \boldsymbol{\rVert}\boldsymbol{v}\boldsymbol{\rVert}^p\boldsymbol{\rVert}\boldsymbol{w}\boldsymbol{\rVert}^p - \boldsymbol{\rangle}\boldsymbol{v},\boldsymbol{w}\boldsymbol{\rangle}^p$，其中$\boldsymbol{v}^p$是将$\boldsymbol{v}$的每个分量取$p$次幂得到的向量。该文献证明此不等式对所有正整数$p$成立，并猜想当$\boldsymbol{v}$和$\boldsymbol{w}$的分量均为正时，该不等式对所有实数$p \boldsymbol{\rceil} 2$成立；还猜想了非负单位向量下该不等式右侧超出左侧的最大量的公式。我们证明了这两个猜想，确定了不等式取等的精确条件，还提出了该不等式针对三个及以上向量的新推广形式。
英文摘要
The following inequality for vectors $\mathbf{v}, \mathbf{w} \in \mathbb{R}^n$ was considered in [Linear and Multilinear Algebra 74:1779--1798 (2026)]: \[\|\mathbf{v}^p\|\|\mathbf{w}^p\| - \langle\mathbf{v}^p,\mathbf{w}^p\rangle \leq \|\mathbf{v}\|^p\|\mathbf{w}\|^p - \langle\mathbf{v},\mathbf{w}\rangle^p,\] where $\mathbf{v}^p$ is the vector that is obtained by raising each entry of $\mathbf{v}$ to the power $p$. It was shown that this inequality holds for all positive integers $p$, and it was conjectured that it holds for all real $p \geq 2$ when the entries of $\mathbf{v}$ and $\mathbf{w}$ are positive. A formula for the maximum amount by which the inequality's right-hand side can exceed its left-hand side for nonnegative unit vectors was also conjectured. We prove both of these conjectures, and we determine exactly when the inequality holds with equality. We also present a new generalization of this inequality to three or more vectors.
Comments14 pages
